In a true relational database, a major table would most likely have related
tables.  when you have related tables and delete the parent record, it will
leave orphans floating around.  If you turn on referential integrity, but do
not allow cascading deleted, it will trow an error if you try to delete the
parent.  If you allow cascading deletes, you run the danger of deleting a
child that belongs to more than one parent.
Now, I could be wrong and you have everything established correctly, but the
evidense is you have only a loose collection of tables.  But even with that,
if another table depends on data in this table, there can be problems.

As to what to do, I don't know.  If the user will not listen to your
professional advise, your only options are to change jobs or comply.  It is
your duty to point out the issues regarding the flaws in the design (as
diplomatically as possible), then do what they want even if it is wrong.  
Then if problems arise, you can cover yourself with the reminder you
suggested a different design.